Euler and skew integration are multiplicative for finite acyclic quivers

Apply [F4] to the small finitary abelian full skeleton supplied by [F3]. Its inclusion preserves isomorphism classes and subobject classes: transporting a monomorphism's domain to its skeleton representative gives a bijection on monomorphisms modulo domain isomorphism, and transports the quotient to the same quotient class. Thus its Hall coefficients are exactly those in [F1]. Write ${\ast }_q$ for this quotient-first product. By [F1], $f\star g=g{\ast }_{q}f$, so $(f\star g)\star h=h{\ast }_q(g{\ast }_{q}f)=(h{\ast }_{q}g){\ast }_{q}f=f\star (g\star h)$ by [F4]. The zero-object class is the unit for ${\ast }_q$, hence also for $\star$.

The degree of an extension is the sum of the endpoint dimension vectors. For a fixed output degree $\mathbf{n}$, a product coefficient uses only the finitely many pairs $\mathbf{d}+\mathbf{e}=\mathbf{n}$; each $H_{Q,\mathbf{d}}$ is finite-dimensional by the representation-matrix definition. Thus [F5] extends the product degree by degree to ${\widehat{H}}_Q$, and truncating above total dimension $D$ changes no coefficient of dimension at most $D$. This is continuity for the total-dimension filtration.

On a normalized basis vector, $I({\kappa }_M)=X^{\mathbf{d}(M)}$ because ${\kappa }_M=\mathrm{\mid }\mathrm{Aut}(M)\mathrm{\mid }{\delta }_M$. For $\mathbf{d}=\mathbf{d}(U)$ and $\mathbf{e}=\mathbf{d}(V)$, [F2] gives $I({\kappa }_U\star {\kappa }_V)=\frac{\mathrm{\mid }{\mathrm{YExt}}^1(V,U)\mathrm{\mid }}{\mathrm{\mid }\mathrm{Hom}(V,U)\mathrm{\mid }}{X}^{\mathbf{d}+\mathbf{e}}\mathrm{.}$ By [F3], both sets are finite vector spaces and the ratio of their cardinalities is $q^{\dim {\mathrm{YExt}}^1(V,U)-\dim \mathrm{Hom}(V,U)}=q^{-{\chi }_Q(\mathbf{e},\mathbf{d})}\mathrm{.}$ This is exactly $I({\kappa }_U)I({\kappa }_V)$ in the Euler target.

On the same basis, $J({\kappa }_M)=t^{{\chi }_Q(\mathbf{d},\mathbf{d})}{x}^{\mathbf{d}}$. Applying [F2] and [F3] to ${\kappa }_U\star {\kappa }_V$ gives its $J$-image coefficient $t^{-2{\chi }_Q(\mathbf{e},\mathbf{d})+{\chi }_Q(\mathbf{d}+\mathbf{e},\mathbf{d}+\mathbf{e})}\mathrm{.}$ By bilinearity, ${\chi }_Q(\mathbf{d}+\mathbf{e},\mathbf{d}+\mathbf{e})={\chi }_Q(\mathbf{d},\mathbf{d})+{\chi }_Q(\mathbf{e},\mathbf{e})+{\chi }_Q(\mathbf{d},\mathbf{e})+{\chi }_Q(\mathbf{e},\mathbf{d})\mathrm{.}$ Since $⟨\mathbf{e},\mathbf{d}⟩={\chi }_Q(\mathbf{e},\mathbf{d})-{\chi }_Q(\mathbf{d},\mathbf{e})$ and $q=t^2$, the coefficient becomes $t^{{\chi }_Q(\mathbf{d},\mathbf{d})+{\chi }_Q(\mathbf{e},\mathbf{e})-⟨\mathbf{e},\mathbf{d}⟩},$ which is exactly the coefficient of $J({\kappa }_U)\ast J({\kappa }_V)$ in the skew target.

Bilinearity of ${\chi }_Q$ makes the Euler twist associative: both parenthesizations of $X^{\mathbf{d}}{X}^{\mathbf{e}}{X}^{\mathbf{f}}$ have exponent $-{\chi }_Q(\mathbf{e},\mathbf{d})-{\chi }_Q(\mathbf{f},\mathbf{d})-{\chi }_Q(\mathbf{f},\mathbf{e})$. Bilinearity of $⟨-,-⟩$ gives the same cocycle identity for the skew product. Both products have the degree-zero basis vector as unit. Step 1.2 gives finite-dimensional homogeneous components, so the formulas for $I$ and $J$ define degree-preserving linear maps and are continuous on the completion. Step 1.1 establishes the source unit and associativity, and steps 1.3–1.4 establish multiplicativity on normalized basis vectors. The cocycle calculation above establishes the target unit and associativity, so linearity and continuity extend $I$ and $J$ to unital algebra homomorphisms on the completions.